A countable regular space without any point of countable character

The Countable Character of Uncountable Graphs

We show that a graph can always be decomposed into edge-disjoint subgraphs of countable cardinality in which the edge-connectivities and edge-separations of the original graph are preserved up to countable cardinals. We also show that the vertex set of any graph can be endowed with a well-ordering which has a certain compactness property with respect to edge-separation.

The Point - Countable Base Problem

Countable subgroups of Euclidean space

In his paper [1], Konstantinos Beros proved a number of results about compactly generated subgroups of Polish groups. Such a group is Kσ, the countable union of compact sets. He notes that the group of rationals under addition with the discrete topology is an example of a Polish group which is Kσ (since it is countable) but not compactly generated (since compact subsets are finite). Beros showe...

Weak Covering without Countable Closure

The main result of [MiSchSt] is that Theorem 0.1 holds under the additional assumption that card(κ) is countably closed. But often, in applications, countable closure is not available. Theorem 0.1 also builds on the earlier covering theorems of Jensen, Dodd and Jensen, and Mitchell; some of the relevant papers are [DeJe], [DoJe1], [DoJe2], [Mi1], [Mi2], and [Je]. The results for smaller core mo...

Co-adaptive learning over a countable space

Co-adaptation is a special form of on-line learning where an algorithm A must assist an unknown algorithm B to perform some task. This is a general framework and has applications in recommendation systems, search, education, and much more. Today, the most common use of co-adaptive algorithms is in brain-computer interfacing (BCI), where algorithms help patients gain and maintain control over pr...